code: def test_cse_mixed_window_functions() -> None:
      # checks if the window caches are cleared
      # there are windows in the cse's and the default expressions
      lf = pl.LazyFrame({"a": [1], "b": [1], "c": [1]})
  
      result = lf.select(
          pl.col("a"),
          pl.col("b"),
          pl.col("c"),
          pl.col("b").rank().alias("rank"),
          pl.col("b").rank().alias("d_rank"),
          pl.col("b").first().over([pl.col("a")]).alias("b_first"),
          pl.col("b").last().over([pl.col("a")]).alias("b_last"),
          pl.col("b").shift().alias("b_lag_1"),
          pl.col("b").shift().alias("b_lead_1"),
          pl.col("c").cum_sum().alias("c_cumsum"),
          pl.col("c")
          .cum_sum()
          .over([pl.col("a")])
          .alias("c_cumsum_by_a"),
          pl.col("c").diff().alias("c_diff"),
          pl.col("c").diff().over([pl.col("a")]).alias("c_diff_by_a"),
      ).collect(comm_subexpr_elim=True)
  
      expected = pl.DataFrame(
          {
              "a": [1],
              "b": [1],
              "c": [1],
              "rank": [1.0],
              "d_rank": [1.0],
              "b_first": [1],
              "b_last": [1],
              "b_lag_1": [None],
              "b_lead_1": [None],
              "c_cumsum": [1],
              "c_cumsum_by_a": [1],
              "c_diff": [None],
              "c_diff_by_a": [None],
          },
      ).with_columns(pl.col(pl.Null).cast(pl.Int64))
      assert_frame_equal(result, expected)
  
imports: import re
  from datetime import date, datetime
  from tempfile import NamedTemporaryFile
  from typing import Any
  import numpy
  import pytest
  import polars
  from polars.testing import assert_frame_equal
label_desc: Use image node_id 10 for calling a global function with example usage: test_cse_mixed_window_functions() without return types